I'm posting this puzzle because there's a very slick trick that allows one to set the first additional digit. Even with that digit set the puzzle is quite challenging. Oh -- this is a new one for me. I've never seen this particular combination (the very slick trick) in a sudoku before. And I didn't spot the new pattern on my own -- Ruud van der Werf had to point it out before I could see it. Thanks for the new idea, Ruud! dcb

I researched the puzzle. Just as well because, left to my own devices, I would not have found the solution in a year. It is certainly an elegant argument. Thinking about it for a couple of days brought a standard alternative approach to mind. An illustration may help those unfamiliar with fins.

Start with an ordinary swordfish based on three rows. The pattern is:

… A … B … C …

… D … E … F …

… G … H … I …

A, B, .. are the nine cells which may admit an entry X, three in each row.

Now imagine cell G replaced by boxG, the box which contains cell G. Thus the bottom row still has only three places for X and they still line up with the rows above. It is just that one of the “places” is three cells lined up in a box rather than a single cell. BoxG is usually called the fin of the fish. The fin can have places for as many X’s as you like.

Logic:
(1) By construction the top two rows between them must contain two X’s.
(2) If either cell A or cell D contains X, X cannot be placed in the intersection of column AD and boxG.
(3) If neither cell A nor cell B contains X, then either cell B and cell F contain X or cell C and cell E contain X. Whichever obtains, neither cell H nor cell I contains X so X must be on row HI and in boxG.
(4) Thus in all cases X can be excluded from two cells in boxG. These cells comprise the intersection of column AD and boxG excluding cell G itself.

There is such a finny swordfish in David’s matrix. X is 7. The top two base rows are row 1 and row 5. The bottom row is row 9 and the fin is box (3, 1), aka box 7. So 7 can be excluded from r7 c2, which must then contain 6, and the difficult step in the puzzle is resolved.

The “fin” argument is quite general. It applies to any m x m fish (m = 2 being an x-wing, m = 3 a swordfish, etc). Also any cell in the fish’s skeleton may be replaced by a box. I do not however think that a useful fish can have two fins.

This standard approach is less elegant than the one David has in mind but it may be more practical. I can just about spot an x-wing in good light but solvers practised at identifying fish may find the fin adds little extra difficulty.

I do not know who to credit with the discovery of the fin and should very much like to if anyone can tell me. Possibly the idea has simply been part of the folklore for some time.